Lefschetz pencil ($f:X⇢{\mathbb{CP}}^1$)

On a closed oriented smooth four-manifold $X$, a Lefschetz pencil is a smooth map $f:X\setminus B\to {\mathbb{CP}}^1$ with finite base locus $B$ and isolated critical points. Orientation-compatible complex local coordinates give the modelsA regular fiber closes to a surface through the base points. Blowing up those points resolves the local ratio and gives a Lefschetz fibration, with an exceptional section for each base point.^[ref]

For a pencil generated by homogeneous polynomials $F_0,F_1$ on a smooth projective surface, $f=[F_0:F_1]$ and $B$ is their common zero set. Generic choices in sufficiently positive linear systems give the isolated complex Morse singularities required here. The local models are stronger conditions than merely having a rational map to the projective line.

Let $X$ be a smooth projective surface over the complex numbers, and let $H$ be an ample divisor. For sufficiently large integers $m$, a general pencil in $|mH|$ is a Lefschetz pencil. Its smooth fiber represents $mH$, so the arithmetic adjunction formula on a smooth surface givesIf $m=4k$, then $m^2{H}^2/2=8k^2{H}^2$ and $m{K}_X\cdot H/2=2k{K}_X\cdot H$ are both even integers. Therefore $g$ is odd. Positivity is needed to obtain a generic Lefschetz pencil; divisibility by four is the separate parity argument. The genericity assertion here uses complex algebraic geometry, rather than an unqualified positive-characteristic version of the complex Morse local model.^[ref]

The standard four-torus has trivial complex tangent bundle, so its symplectic canonical class is zero. The symplectic adjunction formula gives $F^2=2g-2$ for a smooth genus-$g$ fiber of a symplectic Lefschetz pencil. Two generic fibers meet once positively at each base point and nowhere else, giving the displayed count.

A Lefschetz fibration is a proper map from an oriented smooth manifold $X^{2n}$ to an oriented surface, with isolated critical points modeled in orientation-compatible complex coordinates byAway from the critical values it is a proper submersion, hence a smooth fiber bundle by the Ehresmann fibration theorem. Properness is needed for this conclusion; the local critical-point model alone cannot ensure that fibers do not escape to infinity.^[ref]

In real dimension four, a regular fiber is a compact surface. Near a critical value, a circle in that surface shrinks to a node. Blowing up the base points of a Lefschetz pencil supplies a central family of examples: each exceptional sphere becomes a section of the resolved fibration.

If a Lefschetz pencil on a closed four-manifold $X$ has smooth fiber of genus $g$, $b$ base points and $m$ critical points, thenIndeed, blowing up the base points gives $\chi (\tilde{X})=\chi (X)+b$, while a genus-$g$ fibration over $S^2$ contributes $2(2-2g)$ and each Lefschetz critical point contributes one.

Every symplectic smooth fiber in a Lefschetz pencil on ${\mathbb{CP}}^2$ represents $dH$ for an integer $d\ge 1$. The symplectic adjunction formula givesConversely, generic pencils of degree-$d$ complex plane curves realize every value in this list.